Approximate computing is an emerging trend in digital design that trades off the requirement of exact computation for improved speed and power performance. This paper proposes novel approximate compressors and an algorithm to exploit them for the design of efficient approximate multipliers. By using the proposed approach, we have synthesized approximate multipliers for several operand lengths using a 40-nm library. Comparison with previously presented approximated multipliers shows that the proposed circuits provide better power or speed for a target precision. Applications to image filtering and to adaptive least mean squares filtering are also presented in the paper.

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Approximate Multipliers Based on New Approximate Compressors

Approximate computing has emerged as a new paradigm for high-performance and energy-efficient design of circuits and systems. For the many approximate arithmetic circuits proposed, it has become critical to understand a design or approximation technique for a specific application to improve performance and energy efficiency with a minimal loss in accuracy. This article aims to provide a comprehensive survey and a comparative evaluation of recently developed approximate arithmetic circuits under different design constraints. Specifically, approximate adders, multipliers, and dividers are synthesized and characterized under optimizations for performance and area. The error and circuit characteristics are then generalized for different classes of designs. The applications of these circuits in image processing and deep neural networks indicate that the circuits with lower error rates or error biases perform better in simple computations, such as the sum of products, whereas more complex accumulative computations that involve multiple matrix multiplications and convolutions are vulnerable to single-sided errors that lead to a large error bias in the computed result. Such complex computations are more sensitive to errors in addition than those in multiplication, so a larger approximation can be tolerated in multipliers than in adders. The use of approximate arithmetic circuits can improve the quality of image processing and deep learning in addition to the benefits in performance and power consumption for these applications.

Approximate Arithmetic Circuits: A Survey, Characterization, and Recent Applications

Approximate multipliers attract a large interest in the scientific literature that proposes several circuits built with approximate 4-2 compressors. Due to the large number of proposed solutions, the designer who wishes to use an approximate 4-2 compressor is faced with the problem of selecting the right topology. In this paper, we present a comprehensive survey and comparison of approximate 4-2 compressors previously proposed in literature. We present also a novel approximate compressor, so that a total of twelve different approximate 4-2 compressors are analyzed. The investigated circuits are employed to design $8\times 8$ and $16\times 16$ multipliers, implemented in 28nm CMOS technology. For each operand size we analyze two multiplier configurations, with different levels of approximations, both signed and unsigned. Our study highlights that there is no unique winning approximate compressor topology since the best solution depends on the required precision, on the signedness of the multiplier and on the considered error metric.

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Comparison and Extension of Approximate 4-2 Compressors for Low-Power Approximate Multipliers

A scalable approximate multiplier, called truncation- and rounding-based scalable approximate multiplier (TOSAM) is presented, which reduces the number of partial products by truncating each of the input operands based on their leading one-bit position. In the proposed design, multiplication is performed by shift, add, and small fixed-width multiplication operations resulting in large improvements in the energy consumption and area occupation compared to those of the exact multiplier. To improve the total accuracy, input operands of the multiplication part are rounded to the nearest odd number. Because input operands are truncated based on their leading one-bit positions, the accuracy becomes weakly dependent on the width of the input operands and the multiplier becomes scalable. Higher improvements in design parameters (e.g., area and energy consumption) can be achieved as the input operand widths increase. To evaluate the efficiency of the proposed approximate multiplier, its design parameters are compared with those of an exact multiplier and some other recently proposed approximate multipliers. Results reveal that the proposed approximate multiplier with a mean absolute relative error in the range of 11%–0.3% improves delay, area, and energy consumption up to 41%, 90%, and 98%, respectively, compared to those of the exact multiplier. It also outperforms other approximate multipliers in terms of speed, area, and energy consumption. The proposed approximate multiplier has an almost Gaussian error distribution with a near-zero mean value. We exploit it in the structure of a JPEG encoder, sharpening, and classification applications. The results indicate that the quality degradation of the output is negligible. In addition, we suggest an accuracy configurable TOSAM where the energy consumption of the multiplication operation can be adjusted based on the minimum required accuracy.

TOSAM: An Energy-Efficient Truncation- and Rounding-Based Scalable Approximate Multiplier

Exploiting the tradeoff between accuracy and hardware cost has a tremendous potential to improve the efficiency of integrated systems. Using this concept, numerous approximate adders have been proposed in the last ten years. Although conceptually different, all previous architectures have been obtained with an ad hoc and nonsystematic methodology. Instead, this brief generalizes and systematically optimizes an architectural template for approximate adders. The outcome, called optimized lower part constant-OR adder (LOCA), outperforms previous approaches in terms of accuracy and hardware cost. For example, an 8-bit approximate adder implemented with our new approach improves the mean squared error by 58.5%, while simultaneously reducing the cost by 7.2% with respect to the previously reported best architecture.

Systematic Design of an Approximate Adder: The Optimized Lower Part Constant-OR Adder

A variable latency adder (VLA) reduces average addition time by using speculation: the exact arithmetic function is replaced by an approximated one, that is faster and gives correct results most of the times. When speculation fails, an error detection and correction circuit gives the correct result in the following clock cycle. Previous papers investigate VLAs based on Kogge-Stone, Han-Carlson or carry select topologies, speculating that carry propagation involves only a few consecutive bits. In several applications using 2's complement representation, however, operands have a Gaussian distribution and a nontrivial portion of carry chains can be as long as the adder size. In this paper we propose five novel VLA architectures, based on Brent-Kung, Ladner-Fisher, Sklansky, Hybrid Han-Carlson, and Carry increment parallel-prefix topologies. Moreover, we present a new efficient error detection and correction technique, that makes proposed VLAs suitable for applications using 2's complement representation. In order to investigate VLAs performances, proposed architectures have been synthesized using the UMC 65 nm library, for operand lengths ranging from 32 to 128 bits. Obtained results show that proposed VLAs outperform previous speculative architectures and standard (non-speculative) adders when high-speed is required.

Variable Latency Speculative Parallel Prefix Adders for Unsigned and Signed Operands

Sacrificing exact calculations to improve digital circuit performance is at the foundation of approximate computing. In this paper, an approximate multiply-and-accumulate (MAC) unit is introduced. The MAC partial product terms are compressed by using simple OR gates as approximate counters; moreover, to further save energy, selected columns of the partial product terms are not formed. A compensation term is introduced in the proposed MAC, to reduce the overall approximation error. A MAC unit, specialized to perform 2D convolution, is designed following the proposed approach and implemented in TSMC 40nm technology in four different configurations. The proposed circuits achieve power savings more than 60%, compared to standard, exact MAC, with tolerable image quality degradation.

Low-power approximate MAC unit

Variable latency adders have been recently proposed in literature. A variable latency adder employs speculation: the exact arithmetic function is replaced with an approximated one that is faster and gives the correct result most of the time, but not always. The approximated adder is augmented with an error detection network that asserts an error signal when speculation fails. Speculative variable latency adders have attracted strong interest thanks to their capability to reduce average delay compared to traditional architectures. This paper proposes a novel variable latency speculative adder based on Han-Carlson parallel-prefix topology that resulted more effective than variable latency Kogge-Stone topology. The paper describes the stages in which variable latency speculative prefix adders can be subdivided and presents a novel error detection network that reduces error probability compared to previous approaches. Several variable latency speculative adders, for various operand lengths, using both Han-Carlson and Kogge-Stone topology, have been synthesized using the UMC 65 nm library. Obtained results show that proposed variable latency Han-Carlson adder outperforms both previously proposed speculative Kogge-Stone architectures and non-speculative adders, when high-speed is required. It is also shown that non-speculative adders remain the best choice when the speed constraint is relaxed.

Variable Latency Speculative Han-Carlson Adder

Approximate computing improves digital circuit performance by relaxing the requirement of performing exact calculations In this paper, we investigate the use of approximate adders in the final stage of a carry save multiplier-accumulator (MAC), designed for image filtering application We propose a design flow based on synthesis tools, starting from HDL description After a first step in which an exact carry-propagate adder is used, the synthesized netlist is simulated to extract the statistics of the terms summed in the carry-propagate adder We then design the approximate adder, to meet the required error characteristics given the inputs statistics The netlist is finally modified by substituting the exact adder with the approximate one, and a final synthesis and optimization is performed The presented design example in 28nm CMOS shows that a 14% power gain can be obtained, with a limited image quality degradation

Darjn Esposito

Papers

Approximate Multipliers Based on New Approximate Compressors

Variable Latency Speculative Parallel Prefix Adders for Unsigned and Signed Operands

Low-power approximate MAC unit

Variable Latency Speculative Han-Carlson Adder

On the use of approximate adders in carry-save multiplier-accumulators